# On the Failure of Classic Elasticity in Predicting Elastic Wave Propagation in Gyroid Lattices for Very Long Wavelengths

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## Abstract

**:**

## 1. Introduction

#### Notations

- Blackboard fonts will denote tensor spaces: $\mathbb{T}$;
- Tensors of order $>1$ will be denoted using uppercase Roman Bold fonts: $\mathbf{T}$;
- Vectors will be denoted by lowercase Roman Bold fonts: $\mathbf{t}$.

## 2. The Gyroid Lattice

#### 2.1. Parametrization of the Gyroid Lattice

#### 2.2. Symmetry Properties

#### 2.3. Unit Cell

#### 2.3.1. BCC Conventional Unit Cell

#### 2.3.2. BCC Primitive Unit Cell

#### 2.4. Reciprocal Basis and Brillouin Zone

## 3. Analysis Tools

#### 3.1. Bloch–Floquet Analysis

#### 3.2. Polarization of Waves in Homogeneous Materials

## 4. Dispersion Analysis Using Finite Elements Analysis (FEA)

- $[001]$: This direction is going from the center of the fundamental cell to the middle of a face. It corresponds to an axis of rotation of order 4 (rotations of $\pi /2$ rad);
- $[011]$: This direction is going from the center of the fundamental cell to the middle of an edge. It corresponds to an axis of rotation of order 2 (rotations of $\pi $ rad);
- $[111]$: This direction is going from the center of the fundamental cell to a vertex. It corresponds to an axis of rotation of order 3 (rotations of $2\pi /3$ rad).

## 5. Long-Wavelength and Low-Frequency Approximation and Classic Elasticity

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

BCC | Body Centered Cubic |

BZ | Brillouin Zone |

IBZ | Irreducible Brillouin Zone |

FEA | Finite Elements Analysis |

FCC | Face Centered Cubic |

LF | Low Frequency |

LW | Long Wavelength |

## Appendix A. Dictionary

**Table A1.**The set of group generators used to construct matrix representation for each symmetry class.

Group | Generators |
---|---|

$\mathrm{Z}}_{2}^{-$ | $\mathbf{P}}_{{\mathbf{e}}_{3}$ |

$\mathrm{Z}}_{n$ | $\mathbf{R}\left({\mathbf{e}}_{3};\frac{2\pi}{n}\right)$ |

$\mathrm{D}}_{n$ | $\mathbf{R}\left({\mathbf{e}}_{3};\frac{2\pi}{n}\right),\phantom{\rule{4pt}{0ex}}\mathbf{R}({\mathbf{e}}_{1};\pi )$ |

${\mathrm{Z}}_{2n}^{-},\phantom{\rule{4pt}{0ex}}n\ge 2$ | $-\mathbf{R}\left({\mathbf{e}}_{3};\frac{\pi}{n}\right)$ |

${\mathrm{D}}_{2n}^{h}\phantom{\rule{4pt}{0ex}}n\ge 2$ | $-\mathbf{R}\left({\mathbf{e}}_{3};\frac{\pi}{n}\right),\phantom{\rule{4pt}{0ex}}\mathbf{R}({\mathbf{e}}_{1},\pi )$ |

$\mathrm{D}}_{n}^{v$ | $\mathbf{R}\left({\mathbf{e}}_{3};\frac{2\pi}{n}\right),\phantom{\rule{4pt}{0ex}}{\mathbf{P}}_{{\mathbf{e}}_{1}}$ |

$\mathcal{T}$ | $\mathbf{R}({\mathbf{e}}_{3};\pi ),\phantom{\rule{4pt}{0ex}}\mathbf{R}({\mathbf{e}}_{1};\pi ),\phantom{\rule{4pt}{0ex}}\mathbf{R}({\mathbf{e}}_{1}+{\mathbf{e}}_{2}+{\mathbf{e}}_{3};\frac{2\pi}{3})$ |

$\mathcal{O}$ | $\mathbf{R}({\mathbf{e}}_{3};\frac{\pi}{2}),\phantom{\rule{4pt}{0ex}}\mathbf{R}({\mathbf{e}}_{1};\pi ),\phantom{\rule{4pt}{0ex}}\mathbf{R}({\mathbf{e}}_{1}+{\mathbf{e}}_{2}+{\mathbf{e}}_{3};\frac{2\pi}{3})$ |

$\mathcal{O}}^{-$ | $-\mathbf{R}({\mathbf{e}}_{3};\frac{\pi}{2}),\phantom{\rule{4pt}{0ex}}{\mathbf{P}}_{{\mathbf{e}}_{2}-{\mathbf{e}}_{3}}$ |

- $\mathbf{R}(\mathbf{v};\theta )\in \mathrm{SO}(3)$ the rotation about $\mathbf{v}\in {\mathbb{R}}^{3}$ through an angle $\theta \in [0;2\pi ]$;
- ${\mathbf{P}}_{\mathbf{n}}\in \mathrm{O}(3)\setminus \mathrm{SO}(3)$ the reflection through the plane normal to $\mathbf{n}$ (${\mathbf{P}}_{\mathbf{n}}=\mathbf{1}-2\mathbf{n}\otimes \mathbf{n}$).

**Type I Subgroups**

**Table A2.**Dictionary between different group notations for Type I subgroups. The last column indicates the nature of the group: C = Chiral, P = Polar, I = Centrosymetric, and overline indicates that the property is missing.

System | Hermann−Maugin | $\mathbf{Schonflies}$ | $\mathbf{Group}$ | Nature |
---|---|---|---|---|

Triclinic | $1$ | $\mathrm{Z}}_{1$ | $\mathbf{1}$ | $\overline{\mathrm{I}}$CP |

Monoclinic | $2$ | $\mathrm{C}}_{2$ | $\mathrm{Z}}_{2$ | $\overline{\mathrm{I}}$CP |

Orthotropic | $222$ | $\mathrm{D}}_{2$ | $\mathrm{D}}_{2$ | $\overline{\mathrm{I}}$C$\overline{\mathrm{P}}$ |

Trigonal | $3$ | $\mathrm{C}}_{3$ | $\mathrm{Z}}_{3$ | $\overline{\mathrm{I}}$CP |

Trigonal | $32$ | $\mathrm{D}}_{3$ | $\mathrm{D}}_{3$ | $\overline{\mathrm{I}}$C$\overline{\mathrm{P}}$ |

Tetragonal | $4$ | $\mathrm{C}}_{4$ | $\mathrm{Z}}_{4$ | $\overline{\mathrm{I}}$CP |

Tetragonal | $422$ | $\mathrm{D}}_{4$ | $\mathrm{D}}_{4$ | $\overline{\mathrm{I}}$C$\overline{\mathrm{P}}$ |

Hexagonal | $6$ | $\mathrm{C}}_{6$ | $\mathrm{Z}}_{6$ | $\overline{\mathrm{I}}$CP |

Hexagonal | $622$ | $\mathrm{D}}_{6$ | $\mathrm{D}}_{6$ | $\overline{\mathrm{I}}$C$\overline{\mathrm{P}}$ |

$\infty$ | $\mathrm{C}}_{\infty$ | $\mathrm{SO}(2)$ | $\overline{\mathrm{I}}$CP | |

$\infty 2$ | $\mathrm{D}}_{\infty$ | $\mathrm{O}(2)$ | $\overline{\mathrm{I}}$C$\overline{\mathrm{P}}$ | |

Cubic | $23$ | $\mathrm{T}$ | $\mathcal{T}$ | $\overline{\mathrm{I}}$C$\overline{\mathrm{P}}$ |

Cubic | $432$ | $\mathrm{O}$ | $\mathcal{O}$ | $\overline{\mathrm{I}}$C$\overline{\mathrm{P}}$ |

$532$ | $\mathrm{I}$ | $\mathcal{I}$ | $\overline{\mathrm{I}}$C$\overline{\mathrm{P}}$ | |

$\infty \infty$ | $\mathrm{SO}(3)$ | $\overline{\mathrm{I}}$C$\overline{\mathrm{P}}$ |

**Type II Subgroups**

**Table A3.**Dictionary between different group notations for Type II subgroups. The last column indicates the nature of the group: C = Chiral, P = Polar, I = Centrosymetric, and overline indicates that the property is missing.

System | Hermann−Maugin | $\mathbf{Schonflies}$ | $\mathbf{Group}$ | Nature |
---|---|---|---|---|

Triclinic | $\overline{1}$ | $\mathrm{C}}_{i$ | $\mathrm{Z}}_{2}^{c$ | I$\overline{\mathrm{C}}\overline{\mathrm{P}}$ |

Monoclinic | $2/m$ | $\mathrm{C}}_{2h$ | $\mathrm{Z}}_{2}\oplus {\mathrm{Z}}_{2}^{c$ | I$\overline{\mathrm{C}}\overline{\mathrm{P}}$ |

Orthotropic | $mmm$ | $\mathrm{D}}_{2h$ | $\mathrm{D}}_{2}\oplus {\mathrm{Z}}_{2}^{c$ | I$\overline{\mathrm{C}}\overline{\mathrm{P}}$ |

Trigona | $\overline{3}$ | $\mathrm{S}}_{6},\phantom{\rule{4pt}{0ex}}{\mathrm{Z}}_{3i$ | $\mathrm{Z}}_{3}\oplus {\mathrm{Z}}_{2}^{c$ | I$\overline{\mathrm{C}}\overline{\mathrm{P}}$ |

Trigonal | $\overline{3}m$ | $\mathrm{D}}_{3d$ | $\mathrm{D}}_{3}\oplus {\mathrm{Z}}_{2}^{c$ | I$\overline{\mathrm{C}}\overline{\mathrm{P}}$ |

Tetragonal | $4/m$ | $\mathrm{C}}_{4h$ | $\mathrm{Z}}_{4}\oplus {\mathrm{Z}}_{2}^{c$ | I$\overline{\mathrm{C}}\overline{\mathrm{P}}$ |

Tetragonal | $4/mmm$ | $\mathrm{D}}_{4h$ | $\mathrm{D}}_{4}\oplus {\mathrm{Z}}_{2}^{c$ | I$\overline{\mathrm{C}}\overline{\mathrm{P}}$ |

Hexagonal | $6/m$ | $\mathrm{C}}_{6h$ | $\mathrm{Z}}_{6}\oplus {\mathrm{Z}}_{2}^{c$ | I$\overline{\mathrm{C}}\overline{\mathrm{P}}$ |

Hexagonal | $6/mmm$ | $\mathrm{D}}_{6h$ | $\mathrm{D}}_{6}\oplus {\mathrm{Z}}_{2}^{c$ | I$\overline{\mathrm{C}}\overline{\mathrm{P}}$ |

$\infty /m$ | $\mathrm{C}}_{\infty h$ | $\mathrm{SO}(2)\oplus {\mathrm{Z}}_{2}^{c}$ | I$\overline{\mathrm{C}}\overline{\mathrm{P}}$ | |

$\infty /mm$ | $\mathrm{D}}_{\infty h$ | $\mathrm{O}(2)\oplus {\mathrm{Z}}_{2}^{c}$ | I$\overline{\mathrm{C}}\overline{\mathrm{P}}$ | |

Cubic | $m\overline{3}$ | $\mathrm{T}}_{h$ | $\mathcal{T}\oplus {\mathrm{Z}}_{2}^{c}$ | I$\overline{\mathrm{C}}\overline{\mathrm{P}}$ |

Cubic | $m\overline{3}m$ | $\mathrm{O}}_{h$ | $\mathcal{O}\oplus {\mathrm{Z}}_{2}^{c}$ | I$\overline{\mathrm{C}}\overline{\mathrm{P}}$ |

$\overline{5}\overline{3}m$ | $\mathrm{I}}_{h$ | $\mathcal{I}\oplus {\mathrm{Z}}_{2}^{c}$ | I$\overline{\mathrm{C}}\overline{\mathrm{P}}$ | |

$\infty /m\infty /m$ | $\mathrm{O}(3)$ |

**Type III Subgroups**

**Table A4.**Dictionary between different group notations for Type III subgroups. The last column indicates the nature of the group: C = Chiral, P = Polar, I = Centrosymetric, and overline indicates that the property is missing.

System | Hermann−Maugin | $\mathbf{Schonflies}$ | $\mathbf{Group}$ | Nature |
---|---|---|---|---|

Monocinic | $m$ | $\mathrm{C}}_{s$ | $\mathrm{Z}}_{2}^{-$ | $\overline{\mathrm{I}}\overline{\mathrm{C}}$P |

Orthotropic | $2mm$ | $\mathrm{C}}_{2v$ | $\mathrm{D}}_{2}^{v$ | $\overline{\mathrm{I}}\overline{\mathrm{C}}$P |

Trigonal | $3m$ | $\mathrm{C}}_{3v$ | $\mathrm{D}}_{3}^{v$ | $\overline{\mathrm{I}}\overline{\mathrm{C}}$P |

Tetragonal | $\overline{4}$ | $\mathrm{S}}_{4$ | $\mathrm{Z}}_{4}^{-$ | $\overline{\mathrm{I}}\overline{\mathrm{C}}\overline{\mathrm{P}}$ |

Tetragonal | $4mm$ | $\mathrm{C}}_{4v$ | $\mathrm{D}}_{4}^{v$ | $\overline{\mathrm{I}}\overline{\mathrm{C}}$P |

Tetragonal | $\overline{4}2m$ | $\mathrm{D}}_{2d$ | $\mathrm{D}}_{4}^{h$ | $\overline{\mathrm{I}}\overline{\mathrm{C}}\overline{\mathrm{P}}$ |

Hexagonal | $\overline{6}$ | $\mathrm{C}}_{3h$ | $\mathrm{Z}}_{6}^{-$ | $\overline{\mathrm{I}}\overline{\mathrm{C}}\overline{\mathrm{P}}$ |

Hexagonal | $6mm$ | $\mathrm{C}}_{6v$ | $\mathrm{D}}_{6}^{v$ | $\overline{\mathrm{I}}\overline{\mathrm{C}}$P |

Hexagonal | $\overline{6}2m$ | $\mathrm{D}}_{3h$ | $\mathrm{D}}_{6}^{h$ | $\overline{\mathrm{I}}\overline{\mathrm{C}}\overline{\mathrm{P}}$ |

Cubic | $\overline{4}3m$ | $\mathrm{T}}_{d$ | $\mathcal{O}}^{-$ | $\overline{\mathrm{I}}\overline{\mathrm{C}}\overline{\mathrm{P}}$ |

$\infty m$ | $\mathrm{C}}_{\infty v$ | $\mathrm{O}{(2)}^{-}$ | $\overline{\mathrm{I}}\overline{\mathrm{C}}$P |

## Appendix B. Generators of Space Group #214

Seitz | Math | Matrices in Conventional Basis B |
---|---|---|

$\{{2}_{001}|\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\phantom{\rule{4pt}{0ex}}0\phantom{\rule{4pt}{0ex}}\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\}$ | $\left(\mathbf{R}(\pi ,{\mathbf{e}}_{3});\phantom{\rule{4pt}{0ex}}\frac{1}{2}({\mathbf{e}}_{1}+{\mathbf{e}}_{3})\right)$ | ${\left[\begin{array}{cccc}-1& 0& 0& \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\\ 0& -1& 0& 0\\ 0& 0& 1& \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\\ 0& 0& 0& 1\end{array}\right]}_{\mathcal{B}}$ |

$\{{2}_{010}|0\phantom{\rule{4pt}{0ex}}\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\phantom{\rule{4pt}{0ex}}\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\}$ | $\left(\mathbf{R}(\pi ,{\mathbf{e}}_{2});\frac{1}{2}({\mathbf{e}}_{2}+{\mathbf{e}}_{3})\right)$ | ${\left[\begin{array}{cccc}-1& 0& 0& 0\\ 0& 1& 0& \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\\ 0& 0& -1& \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\\ 0& 0& 0& 1\end{array}\right]}_{\mathcal{B}}$ |

$\{{3}_{111}^{+}|0\}$ | $\left(\mathbf{R}(\frac{2\pi}{3};\phantom{\rule{4pt}{0ex}}{\mathbf{e}}_{1}+{\mathbf{e}}_{2}+{\mathbf{e}}_{3}),\underline{0}\right)$ | ${\left[\begin{array}{cccc}0& 0& 1& 0\\ 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\end{array}\right]}_{\mathcal{B}}$ |

$\{{2}_{110}|\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\phantom{\rule{4pt}{0ex}}\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\phantom{\rule{4pt}{0ex}}\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\}$ | $\left(\mathbf{R}(\pi ,{\mathbf{e}}_{1}+{\mathbf{e}}_{2});\phantom{\rule{4pt}{0ex}}\frac{1}{4}(3{\mathbf{e}}_{1}+{\mathbf{e}}_{2}+{\mathbf{e}}_{3})\right)$ | ${\left[\begin{array}{cccc}0& 1& 0& \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\\ 1& 0& 0& \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\\ 0& 0& -1& \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\\ 0& 0& 0& 1\end{array}\right]}_{\mathcal{B}}$ |

$\{1|\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\phantom{\rule{4pt}{0ex}}\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\phantom{\rule{4pt}{0ex}}\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\}$ | $\left(\mathrm{Id};\phantom{\rule{4pt}{0ex}}\frac{1}{2}({\mathbf{e}}_{1}+{\mathbf{e}}_{2}+{\mathbf{e}}_{3})\right)$ | ${\left[\begin{array}{cccc}1& 0& 0& \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\\ 0& 1& 0& \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\\ 0& 0& 1& \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\\ 0& 0& 0& 1\end{array}\right]}_{\mathcal{B}}$ |

## Appendix C. Proof

**Figure A1.**(

**a**) The gyroid restricted to its fundamental domain along with symmetry axes ${\mathrm{C}}_{3}$ (plain) and ${\mathrm{C}}_{2}$ (dashed) and (

**b**) evolution of parameter b as a function of x position along ${\mathbf{e}}_{1}$ axis, dashed line corresponds to $b=\sqrt{2}$.

- -
- Rotation of angle $\raisebox{1ex}{$2\pi $}\!\left/ \!\raisebox{-1ex}{$3$}\right.$ along the axis defined by equations $y=z=x$, plotted in plain line in Figure A1 and corresponding to the transformation $(x,y,z)\to (y,z,x)$. The directing vector of this axis is $(1,1,1)$ in orthonormal basis $\mathcal{B}$ and passes through point $(0,0,0)$;
- -
- Three rotations of angle $\pi $ about the three axes defined by equations $\{y=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.-x,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}z=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$8$}\right.\}$, $\{z=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.-x,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}y=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$8$}\right.\}$, and $\{z=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.-y,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}x=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$8$}\right.\}$ and plotted in dashed lines in Figure A1. These axes correspond to transformations $(x,y,z)\to (\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.-y,\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.-x,\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.-z)$, $(x,y,z)\to (\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.-z,\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.-y,\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.-x)$, $(x,y,z)\to (\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.-x,\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.-z,\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.-y)$, respectively. The directing vector of these axes are, in orthonormal basis $\mathcal{B}$, $(1,-1,0)$, $(1,0,-1)$ and $(0,1,-1)$ and they pass through points $(0,\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.,\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$8$}\right.)$, $(0,\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$8$}\right.,\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.)$ and $(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$8$}\right.,0,\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.)$, respectively.

**Table A6.**The expression of the normal director to the gyroid surface at the intersection point with its symmetry axes and expression of this intersection point.

Sym. axis & Directing Vector | Normal to the Gyroid Surface | Intersection Point |
---|---|---|

${\mathrm{C}}_{3}$${(1,1,1)}_{\mathcal{B}}$ | $\left({cos}^{2}2\pi x-{sin}^{2}2\pi x\right){(1,1,1)}_{\mathcal{B}}$ | $(x,x,x)$ with $3cos2\pi xsin2\pi x=b$ |

${\mathrm{C}}_{2}$${(1,-1,0)}_{\mathcal{B}}$ | $sin2\pi x\left(\raisebox{1ex}{$\sqrt{2}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.-cos2\pi x\right){(1,-1,0)}_{\mathcal{B}}$ | $(x,-x,0)$ with $\sqrt{2}cos2\pi x+{sin}^{2}2\pi x=b$ |

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**Figure 1.**The unit cells of gyroid lattices obtained for $a=1$ mm and: (

**a**) $b=0$, (

**b**) $b=1$, (

**c**) $b=1.3$. Despite what the angle of view may suggest, all these structures are simply connected.

**Figure 2.**The relationship between the parameter b and the porosity, dashed line represents a linear approximation of real porosity plotted in plain line. The evaluation of the porosity is obtained numerically.

**Figure 4.**Two examples of Primitive Unit Cells (PUC). In (

**a**,

**b**) the Conventional Unit Cell are in blue and the PUC are in red. The PUC in (

**a**) is considered in the present paper, as their lattice vectors (indicated in red) are more symmetrical than those of the one in (

**b**).

**Figure 5.**Some examples of polarizations: (

**a**) Linear, (

**b**) circular right handed, and (

**c**) circular left handed.

**Figure 7.**The dispersion diagram of the Gyroid lattice computed along the boundaries of the Irreducible Brillouin Zone (IBZ) and directions of propagation with respect to the unit cell. (

**a**) The dispersion relation of the Gyroid lattice computed along the boundaries of the IBZ. (

**b**) The direction of propagation with respect to the unit cell.

**Figure 8.**The photonic band diagram of a single gyroid photonic crystal. Reproduced with permission from [6].

**Figure 9.**The phase velocity of the circularly polarized waves in function of the wavenumber or of the ratio between the wavelength $\lambda $ and the size of the unit cell a for propagation direction $[001]$ (

**a**) and $[111]$ (

**b**). Right handed waves are in black, left handed waves are in gray. The dashed horizontal lines correspond to the phase velocity to which they converge to for infinite wavelength.

**Figure 10.**An illustration of circular birefringence observed along direction [100] at ${f}_{c}=200$ kHz, corresponding to a wavelength around 10 times the size of the unit cell. A linearly polarized wave entering the material is subjected to a rotation of 1.3 degrees/wavelength, then after 20 wavelengths the rotation is ${\theta}_{t}=26.0$ deg. This illustration does not account for the changes in amplitude due to the reflections at boundaries.

**Table 1.**The high symmetry points of the gyroid lattice. The group notations are detailed in Appendix A.

Symmetry | Coordinates | Coordinates | Point Group | Point Group | Illustration of the |
---|---|---|---|---|---|

Point | w.r.t. ${\mathcal{P}}^{\u2b51}$ | w.r.t. $\mathcal{B}$ | (Math.) | (H-M) | First Brillouin Zone |

$({k}_{1},{k}_{2},{k}_{3})$ | $({x}_{1},{x}_{2},{x}_{3})$ | ||||

$\mathsf{\Gamma}$ | $(0,0,0)$ | $(0,0,0)$ | $\mathcal{O}$ | 432 | |

H | $(-\frac{1}{2},\frac{1}{2},\frac{1}{2})$ | $(0,0,\frac{1}{a})$ | $\mathcal{O}$ | 432 | |

P | $(\frac{1}{4},\frac{1}{4},\frac{1}{4})$ | $(\frac{1}{2a},\frac{1}{2a},\frac{1}{2a})$ | ${\mathrm{D}}_{3}$ | 32 | |

N | $(0,\frac{1}{2},0)$ | $(0,\frac{1}{2a},\frac{1}{2a})$ | ${\mathrm{D}}_{2}$ | 222 |

Polarization | Condition |
---|---|

Longitudinal polarization | $\mathbf{U}=\alpha \mathbf{n}$ with $\alpha \in \mathbb{C}$ |

Transverse polarization | ${\mathbf{U}}^{\mathbb{R}}$ and ${\mathbf{U}}^{\mathbb{C}}$ belong to the plane orthogonal to $\mathbf{n}$ |

Linear polarization | $\mathbf{U}\wedge {\mathbf{U}}^{*}=\mathbf{0}$, with ${\mathbf{U}}^{*}$ complex conjugate of $\mathbf{U}$ |

Circular polarization | $\mathbf{U}\xb7\mathbf{U}=0$ |

↳ Right handedness | $\mathbf{n}\xb7({\mathbf{U}}^{\mathbb{R}}\wedge {\mathbf{U}}^{\mathbb{C}})<0$ |

↳ Left handedness | $\mathbf{n}\xb7({\mathbf{U}}^{\mathbb{R}}\wedge {\mathbf{U}}^{\mathbb{C}})>0$ |

Elliptic polarization | $\mathbf{U}\wedge {\mathbf{U}}^{*}\ne \mathbf{0}$ and $\mathbf{U}\xb7\mathbf{U}\ne 0$ |

Mass Density [kg/m${}^{2}$] | Young Modulus [GPa] | Poisson Ratio [1] | Porosity [1] | Unit Cell Size [mm] |
---|---|---|---|---|

${\mathit{\rho}}_{\mathit{b}}$ | ${\mathit{E}}_{\mathit{b}}$ | ${\mathit{\nu}}_{\mathit{b}}$ | $\mathit{p}$ | $\mathit{a}$ |

4506 | 115.7 | 0.321 | 0.7 | 1 |

Direction | Phase Velocity | Polarization | Type of Wave |
---|---|---|---|

[m/s] | $({x}_{1},{x}_{2},{x}_{3})$ | ||

$3018.5$ | $(0,0,1)$ | Longitudinal | |

$[001]$ | $1933.4$ | $(1,i,0)$ | Circular L ⥁ |

$1931.0$ | $(1,-i,0)$ | Circular R ⥀ | |

3249.8 | $(0,1,1)$ | Longitudinal | |

$[011]$ | 1931.7 | $(1,0,0)$ | Transverse |

1510.9 | $(0,1,-1)$ | Transverse | |

3322.8 | $(1,1,1)$ | Longitudinal | |

$[111]$ | 1664.6 | $(1,-0.54-i0.88,-0.46+i0.88)$ | Circular R ⥀ |

1662.3 | $(1,-0.45+i0.85,-0.55-i0.85)$ | Circular L ⥁ |

Direction | Phase Velocities [m/s] | Polarization | Type of Wave |
---|---|---|---|

$\sqrt{\frac{{c}_{11}}{\rho}}\approx 3018.5$ | $(0,0,1)$ | Longitudinal | |

$[001]$ | $\sqrt{\frac{{c}_{44}}{2\rho}}\approx 1932.2$ | $(0,1,0)$ | Transverse |

$\sqrt{\frac{{c}_{44}}{2\rho}}\approx 1932.2$ | $(1,0,0)$ | Transverse | |

$\sqrt{\frac{{c}_{11}+{c}_{12}+{c}_{44}}{2\rho}}\approx 3249.8$ | $(0,1,1)$ | Longitudinal | |

$[011]$ | $\sqrt{\frac{{c}_{44}}{2\rho}}\approx 1932.2$ | $(1,0,0)$ | Transverse |

$\sqrt{\frac{{c}_{11}-{c}_{12}}{2\rho}}\approx 1510.9$ | $(0,-1,1)$ | Transverse | |

$\sqrt{\frac{{c}_{11}+2{c}_{12}+2{c}_{44}}{3\rho}}\approx 3322.8$ | $(1,1,1)$ | Longitudinal | |

$[111]$ | $\sqrt{\frac{2{c}_{11}-2{c}_{12}+{c}_{44}}{6\rho}}\approx 1663.4$ | $(-1,1,0)$ | Transverse |

$\sqrt{\frac{2{c}_{11}-2{c}_{12}+{c}_{44}}{6\rho}}\approx 1663.4$ | $(-1,-1,2)$ | Transverse |

Elastic Coefficients | Mass Density | ||
---|---|---|---|

${\mathit{c}}_{11}$ [GPa] | ${\mathit{c}}_{12}$ [GPa] | ${\mathit{c}}_{44}=2{\mathit{c}}_{2323}$ [GPa] | $\mathit{\rho}$ [kg/m${}^{3}$] |

12.32 | 6.145 | 10.092 | 1351.8 |

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**MDPI and ACS Style**

Rosi, G.; Auffray, N.; Combescure, C.
On the Failure of Classic Elasticity in Predicting Elastic Wave Propagation in Gyroid Lattices for Very Long Wavelengths. *Symmetry* **2020**, *12*, 1243.
https://doi.org/10.3390/sym12081243

**AMA Style**

Rosi G, Auffray N, Combescure C.
On the Failure of Classic Elasticity in Predicting Elastic Wave Propagation in Gyroid Lattices for Very Long Wavelengths. *Symmetry*. 2020; 12(8):1243.
https://doi.org/10.3390/sym12081243

**Chicago/Turabian Style**

Rosi, Giuseppe, Nicolas Auffray, and Christelle Combescure.
2020. "On the Failure of Classic Elasticity in Predicting Elastic Wave Propagation in Gyroid Lattices for Very Long Wavelengths" *Symmetry* 12, no. 8: 1243.
https://doi.org/10.3390/sym12081243